Estimation of a $k$-monotone density: limit distribution theory and the spline connection

Details Estimation of a $k$-monotone density: limit distribution theory and the spline connection Estimation of a $k$-monotone density: limit distribution theory and the spline connection

2005-09-04

arxiv.org/abs/math/0509081v2

Annals of Statistics 2007, Vol. 35, No. 6, 2536-2564

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/009053607000000262 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000262

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We find that the $j$th derivative of the estimators at $x_0$ converges at the rate $n^{-(k-j)/(2k+1)}$ for $j=0,...,k-1$. The limiting distribution depends on an almost surely uniquely defined stochastic process $H_k$ that stays above (below) the $k$-fold integral of Brownian motion plus a deterministic drift when $k$ is even (odd). Both the MLE and LSE are known to be splines of degree $k-1$ with simple knots. Establishing the order of the random gap $\tau_n^+-\tau_n^-$, where $\tau_n^{\pm}$ denote two successive knots, is a key ingredient of the proof of the main results. We show that this ``gap problem'' can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.

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